© 2004 Nature Publishing Group http://www.nature.com/naturebiotechnology
_computational
BIOLOGY
PRIMER
What is a hidden Markov model?
Sean R Eddy
Statistical models called hidden Markov models are a recurring theme in computational biology. What are hidden
Markov models, and why are they so useful for so many different problems?
Often, biological sequence analysis is just a
matter of putting the right label on each
residue. In gene identification, we want to
label nucleotides as exons, introns, or intergenic sequence. In sequence alignment, we
want to associate residues in a query
sequence with homologous residues in a target database sequence. We can always write
an ad hoc program for any given problem,
but the same frustrating issues will always
recur. One is that we want to incorporate
heterogeneous sources of information. A
genefinder, for instance, ought to combine
splice-site consensus, codon bias, exon/
intron length preferences and open reading
frame analysis into one scoring system. How
should these parameters be set? How should
different kinds of information be weighted?
A second issue is to interpret results probabilistically. Finding a best scoring answer is
one thing, but what does the score mean,
and how confident are we that the best scoring answer is correct? A third issue is extensibility. The moment we perfect our ad hoc
genefinder, we wish we had also modeled
translational initiation consensus, alternative splicing and a polyadenylation signal.
Too often, piling more reality onto a fragile
ad hoc program makes it collapse under its
own weight.
Hidden Markov models (HMMs) are a
formal foundation for making probabilistic
models of linear sequence ‘labeling’ problems1,2. They provide a conceptual toolkit
for building complex models just by draw-
Sean R. Eddy is at Howard Hughes Medical
Institute & Department of Genetics,
Washington University School of Medicine,
4444 Forest Park Blvd., Box 8510, Saint Louis,
Missouri 63108, USA.
e-mail: [email protected]
A = 0.25
C = 0.25
G = 0.25
T = 0.25
Start
1.0
E
0.1
A = 0.05
C=0
G = 0.95
T =0
5
A = 0.4
C = 0.1
G = 0.1
T = 0.4
1.0
0.9
Sequence:
State path:
End
I
0.1
0.9
C T T C A TG TG A A AG C AG AC G T A A G T C A
EE EEEEEEEEEEEEEEEE 5 I I I I I I I
log P
–41.22
–43.90
–43.45
–43.94
–42.58
–41.71
Parsing:
46%
Posterior
decoding:
28%
11%
Figure 1 A toy HMM for 5′ splice site recognition. See text for explanation.
ing an intuitive picture. They are at the heart
of a diverse range of programs, including
genefinding, profile searches, multiple
sequence alignment and regulatory site
identification. HMMs are the Legos of computational sequence analysis.
A toy HMM: 5′ splice site recognition
As a simple example, imagine the following
caricature of a 5′ splice-site recognition
problem. Assume we are given a DNA
sequence that begins in an exon, contains
one 5′ splice site and ends in an intron.
The problem is to identify where the switch
from exon to intron occurred—where the
5′ splice site (5′SS) is.
For us to guess intelligently, the sequences
of exons, splice sites and introns must have
NATURE BIOTECHNOLOGY VOLUME 22 NUMBER 10 OCTOBER 2004
different statistical properties. Let’s imagine
some simple differences: say that exons
have a uniform base composition on average
(25% each base), introns are A/T rich (say,
40% each for A/T, 10% each for C/G), and
the 5′SS consensus nucleotide is almost
always a G (say, 95% G and 5% A).
Starting from this information, we can
draw an HMM (Fig. 1). The HMM invokes
three states, one for each of the three labels
we might assign to a nucleotide: E (exon),
5 (5′SS) and I (intron). Each state has its
own emission probabilities (shown above the
states), which model the base composition
of exons, introns and the consensus G at the
5′SS. Each state also has transition probabilities (arrows), the probabilities of moving
from this state to a new state. The transition
1315
PRIMER
© 2004 Nature Publishing Group http://www.nature.com/naturebiotechnology
probabilities describe the linear order in
which we expect the states to occur: one or
more Es, one 5, one or more Is.
So, what’s hidden?
It’s useful to imagine an HMM generating a
sequence. When we visit a state, we emit a
residue from the state’s emission probability
distribution. Then, we choose which state to
visit next according to the state’s transition
probability distribution. The model thus
generates two strings of information. One
is the underlying state path (the labels), as
we transition from state to state. The other
is the observed sequence (the DNA), each
residue being emitted from one state in the
state path.
The state path is a Markov chain, meaning
that what state we go to next depends only
on what state we’re in. Since we’re only given
the observed sequence, this underlying state
path is hidden—these are the residue labels
that we’d like to infer. The state path is a
hidden Markov chain.
The probability P(S,π|HMM,θ) that an
HMM with parameters θ generates a state
path π and an observed sequence S is the
product of all the emission probabilities and
transition probabilities that were used.
For example, consider the 26-nucleotide
sequence and state path in the middle of
Figure 1, where there are 27 transitions and
26 emissions to tote up. Multiply all 53
probabilities together (and take the log,
since these are small numbers) and you’ll
calculate log P(S,π|HMM,θ) = –41.22.
An HMM is a full probabilistic model —
the model parameters and the overall
sequence ‘scores’ are all probabilities. Therefore, we can use Bayesian probability theory
to manipulate these numbers in standard,
powerful ways, including optimizing parameters and interpreting the significance
of scores.
Finding the best state path
In an analysis problem, we’re given a
sequence, and we want to infer the hidden
state path. There are potentially many state
paths that could generate the same
sequence. We want to find the one with the
highest probability.
For example, if we were given the HMM
and the 26-nucleotide sequence in Figure 1,
there are 14 possible paths that have nonzero probability, since the 5′SS must fall on
one of 14 internal As or Gs. Figure 1 enumerates the six highest-scoring paths (those
1316
with G at the 5′SS). The best one has a log
probability of –41.22, which infers that the
most likely 5′SS position is at the fifth G.
For most problems, there are so many
possible state sequences that we could not
afford to enumerate them. The efficient
Viterbi algorithm is guaranteed to find the
most probable state path given a sequence
and an HMM. The Viterbi algorithm is a
dynamic programming algorithm quite
similar to those used for standard sequence
alignment.
Beyond best scoring alignments
Figure 1 shows that one alternative state
path differs only slightly in score from putting the 5′SS at the fifth G (log probabilities
of –41.71 versus –41.22). How confident are
we that the fifth G is the right choice?
This is an example of an advantage of
probabilistic modeling: we can calculate our
confidence directly. The probability that
residue i was emitted by state k is the sum
of the probabilities of all the state paths
that use state k to generate residue i (that is,
πi = k in the state path π), normalized by the
sum over all possible state paths. In our toy
model, this is just one state path in the
numerator and a sum over 14 state paths in
the denominator. We get a probability of
46% that the best-scoring fifth G is correct
and 28% that the sixth G position is correct
(Fig. 1, bottom). This is called posterior
decoding. For larger problems, posterior
decoding uses two dynamic programming
algorithms called Forward and Backward,
which are essentially like Viterbi, but they
sum over possible paths instead of choosing
the best.
Making more realistic models
Making an HMM means specifying four
things: (i) the symbol alphabet, K different
symbols (e.g., ACGT, K = 4); (ii) the number
of states in the model, M; (iii) emission
probabilities ei(x) for each state i, that sum
to one over K symbols x, Σ ei(x) = 1; and
x
(iv) transition probabilities ti(j) for each state
i going to any other state j (including itself)
that sum to one over the M states j, Σ ti(j) = 1.
For example, in our toy splice-site model,
maybe we’re not happy with our discrimination power; maybe we want to add a more
realistic six-nucleotide consensus GTRAGT
at the 5′ splice site. We can put a row of
six HMM states in place of ‘5’ state, to
model a six-base ungapped consensus motif,
parameterizing the emission probabilities
on known 5′ splice sites. And maybe we
want to model a complete intron, including
a 3′ splice site; we just add a row of states for
the 3′SS consensus, and add a 3′ exon state
to let the observed sequence end in an exon
instead of an intron. Then maybe we want to
build a complete gene model…whatever
we add, it’s just a matter of drawing what
we want.
The catch
HMMs don’t deal well with correlations
between residues, because they assume that
each residue depends only on one underlying state. An example where HMMs are
usually inappropriate is RNA secondary
structure analysis. Conserved RNA base
pairs induce long-range pairwise correlations; one position might be any residue,
but the base-paired partner must be complementary. An HMM state path has no
way of ‘remembering’ what a distant state
generated.
Sometimes, one can bend the rules of
HMMs without breaking the algorithms.
For instance, in genefinding, one wants to
emit a correlated triplet codon instead of
three independent residues; HMM algorithms can readily be extended to tripletemitting states. However, the basic HMM
toolkit can only be stretched so far. Beyond
HMMs, there are more powerful (though
less efficient) classes of probabilistic models
for sequence analysis.
1. Rabiner, L.R. A tutorial on hidden Markov models and
selected applications in speech recognition. Proc.
IEEE 77, 257–286 (1989).
2. Durbin, R., Eddy, S.R., Krogh, A. & Mitchison, G.J.
Biological Sequence Analysis: Probabilistic Models of
Proteins and Nucleic Acids (Cambridge University
Press, Cambridge UK, 1998).
j
Any model that has these properties is an HMM.
This means that one can make a new
HMM just by drawing a picture corresponding to the problem at hand, like
Figure 1. This graphical simplicity lets one
focus clearly on the biological definition of
a problem.
Wondering how some other
mathematical technique really works?
Send suggestions for future primers to
[email protected].
VOLUME 22 NUMBER 10 OCTOBER 2004 NATURE BIOTECHNOLOGY